Example of Normal Distribution

Problem Statement

Let's consider human heights as an example of a normal distribution.

Example Data

Suppose the heights of adult males in a population follow a normal distribution with:

• Mean (μ) = 175 cm
• Standard deviation (σ) = 10 cm

Using this, we can calculate probabilities, percentiles, and visualize the distribution.

1. Probability Calculation

Q: What is the probability that a randomly chosen person is taller than 185 cm?

Using the Z-score formula:

Z = (X - μ) / σ
Z = (185 - 175) / 10 = 1.0
    

From the Z-table, the probability of a value being less than 185 cm is 0.8413.

Thus, the probability of a height greater than 185 cm:

P(X > 185) = 1 - 0.8413 = 0.1587
    

So, 15.87% of people are taller than 185 cm.

2. Visual Representation

The normal distribution curve for this example would look like:

📈 Bell Curve Representation

• 68% of people fall between 165 cm and 185 cm (±1σ).
• 95% fall between 155 cm and 195 cm (±2σ).
• 99.7% fall between 145 cm and 205 cm (±3σ).

3. Real-World Applications in AI

• Gaussian Naive Bayes classifier assumes normality of feature distributions.
• Anomaly detection identifies data points deviating significantly from the normal curve.
• Feature scaling & transformation in ML assumes normality for many algorithms.